Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 2

Use a concurrent comparison: test the same specimens for both cocaine and mephedrone, rather than treating the historical cocaine percentage as an exactly known 2010 parameter. Assume representative sampling of privates, effectively independent specimens, stable underlying rates during the sampling period, and sufficiently accurate assays. I use a two-sided 5% test of equal positive-test probabilities, with 80% statistical power at $p_C=0.010$ and $p_M=0.005$. This is a design to distinguish the rates at a twofold alternative; a significance test does not prove that their ratio is exactly two.

The paired design needs the joint positive-test distribution. For specimen $i$, write $C_i,M_i\in \{0,1\}$ and $D_i=C_i-M_i$. Put $q_{11}=P(C_i=M_i=1)$ andThen $\mathbb{E}{D}_i=\delta$ and $\mathrm{Var}(D_i)=q-{\delta }^2$. Under equal marginal rates, the two discordant outcomes have equal probabilities; conditional on the total number of discordant specimens, their split is a fair binomial distribution. This gives the McNemar test. Its large-sample version rejects for a sufficiently large absolute difference between the discordant counts, divided by their estimated standard deviation.

For planning, suppose simultaneous positives are negligible, so $q\simeq 0.015$. Under the alternative the difference in positive counts has mean $n\delta$ and standard deviation $\sqrt{n(q-{\delta }^2)}$; the approximate upper null rejection boundary is $z_{0.975}\sqrt{nq}$. Requiring 80% upper-tail rejection probability gives the paired binary sample size calculationAdd the mephedrone assay to about 5,000 existing tests, roughly ten weeks at 500 per week. An exact conditional-power calculation under the negligible-overlap model gives about $77.6\%$ at 4,707 specimens and $80.2\%$ at 5,000, so the rounding also compensates for the conservative discrete test. If positivity for the two drugs were independent within a specimen, $q_{11}=0.00005$, giving a very similar requirement of about 4,676. For the stated marginal rates, $q\le 0.015$, so negligible overlap is a conservative variance choice within this approximation. Substantial co-use reduces the discordant proportion and changes the necessary size. Repeated tests on the same person, unit-level clustered data or inaccurate assays can instead reduce effective information and require more specimens.

A pre-specified one-sided 5% comparison would use $1.645$ in place of $1.96$ and require about 3,708 specimens before allowing for discreteness under the negligible-overlap assumption. These are different testing conventions. Treating the historical 1% as fixed would give a one-sample calculation, but would ignore uncertainty or changes in the actual 2010 cocaine rate.

Model deaths in disjoint equal-duration periods as independent observations with Poisson distributions with comparable population exposure and ascertainment. The expected before and after counts are ${\mu }_0=400$ and ${\mu }_1=320$. Their difference hasThe expected signal relative to its standard deviation is thereforeUnder the null of equal rates, condition on $N=D_0+D_1$: the first-period count has a binomial distribution with parameters $N$ and $1/2$. At a total near 720, the two-sided 5% rejection boundary for $D_0-D_1$ is approximately $1.96\sqrt{720}$. Under the proposed alternative, conditional on that total, its mean is $N/9$ and its variance is $80N/81$. Thus at $N=720$ the normal approximation gives power approximatelyThe lower-tail rejection probability is negligible at this alternative; allowing the Poisson total to fluctuate and using discrete rejection cutoffs changes the approximation slightly: an exact two-sided conditional test has about $83.8\%$ power under these Poisson means. Two years on each side should have adequate power for a 20% reduction under this model, though a particular realization can still fail to detect it.

This conclusion needs stable recording of drug-related deaths, comparable exposure denominators and no substantial overdispersion or temporal dependence. A change in population size, certification practice or other drug use can affect the comparison. Detecting a decline does not establish that mephedrone caused it: attributing displacement requires a credible comparison of potential outcomes, using an otherwise expected trend or other controls for changes occurring at the same time.

Expected death counts are the number of users multiplied by the death risk per user, with matching time windows and definitions. Relative to the MDMA benchmark, the proposed use prevalence contributes a multiplier of $0.6$ and the per-user fatality risk contributes a multiplier of $0.5$. HenceThis assumes comparable user exposure durations and recording of single-drug deaths. Population death counts alone would not identify a per-user risk ratio without the prevalence denominator.

On the stipulated figures, a 20% decline from 200 cocaine-related deaths means about 40 fewer deaths, against about 10 attributed to mephedrone:That bookkeeping is relevant to a population-harm assessment only if the avoided deaths really result from substitution, the new deaths are counted comparably, and overlapping drug attributions are not counted twice. A change in the legality of one drug is an intervention affecting behaviour; the before-and-after association alone does not identify its causal effect.

Distinguish per-user danger from total population harm. More use can raise total deaths even for a less dangerous drug, and an increase can involve new users who would otherwise use no drug. Conversely, restricting a substitute may move users back to more dangerous alternatives. The relative fatality risks do not mechanically determine legal classification: nonfatal harm, dependence, dosage and purity, co-use, the composition of users, enforcement and supply effects, and uncertainty in the estimates all matter. Any policy assessment should compare plausible behavioural responses to the alternatives, including displacement of both cocaine and MDMA, rather than extrapolate from ten deaths alone.